University Brunswick Aerotriangulation

Prévia do material em texto

AEROTRIANGULATION
W. FAIG
June 1976
TECHNICAL REPORT 
NO. 217
LECTURE NOTES
NO. 40
 
 
 
AEROTRIANGULATION 
 
 
 
 
 
 
W. Faig 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Department of Geodesy and Geomatics Engineering 
University of New Brunswick 
P.O. Box 4400 
Fredericton, N.B. 
Canada 
E3B 5A3 
 
 
 
 
 
June 1976 
 
 
PREFACE 
 
In order to make our extensive series of lecture notes more readily available, we have 
scanned the old master copies and produced electronic versions in Portable Document 
Format. The quality of the images varies depending on the quality of the originals. The 
images have not been converted to searchable text. 
FORWARD 
These lecture notes have been written to support the lectures 
in aerotriangulation at UNB. They should be regarded just as such and 
can by no means be considered complete. For most approaches examples 
have been selected while many other formulations are mentioned or even 
neglected. This selection was based partly on the availability of 
programmes to our students and is not intended as classification. 
i 
W. Faig 
Summer 1976 
1. 
TABLE OF CONTENTS 
Introduction ........ . 
1.1 Purpose and Definition of AeroTriangulation .. 
1.2 Applications of Aerotriangulation ..... . 
. 1 
. . 1 
• . 1 
1.3 Some Remarks Concerning the History of Aerotriangulation ... 2 
1.4 Overview of Methods of Aerotriangulation . . . . . 2 
1.5 Some Remarks on Basic Analytical Photogrammetry. . . . n 
2. Graphical and Mechanical Methods of Aerotriangulation . . . . . 8 
2.1 Basic Idea of Radial Triangulation ... . 
2.2 Methods of Radial Triangulation ... . 
2.2.1 Graphical Method ......... . 
2.2.2 Numerical Methods of Radial Triangulation 
2.2.3 Slotted Templets . 
2.2.4 Stereo Templets 
. . 8 
• • 1 0 
. . 10 
. .1 0 
.10 
. . 11 
2.2.5 Jerie•s ITC Analogue Computer . . . . . .... 12 
3 Methdds of Strtp Triangulation and Instrumentation . .14 
3.1 The Principle of Continuous Cantilever Extension with Scale 
Transfer . . . . . . . . . 
3.2 Triangulation Instruments ..... 
3.3 Strip Triangulation with Plotters without Base Change 
3.4 Precision Comparators .............. . 
3.5 Point Selection, Transfer, Marking, Targetting etc .. 
4. Errot Theory of Strip Triangulation .......... . 
4.1 General Assumptions ............. . 
4.2 Vermeir•s Simplified Theory of Transfer Errors .. 
4.3 Coordinate Errors in Strip Axis 
.. 14 
. .. 16 
.. . 20 
.21 
. . . 31 
• • • • • • 37 
. .. 37 
. .37 
.. 40 
4.4 Double Summation ....... . • . . • • • . . . . 42 
ii 
4.5 Off-Axis Points and Their Coordinate Errors ......... 44 
5. Strip Adjustment with Polynomials . . . . . . . . . . . . 51 
5.1 General Remarks . . . . . . . . . . . . . . . . .... 51 
5.2 The U.S. Coast and Geodetic Survey Method, as Example . . 54 
6. Aerotriangulation with Independent Models . . . 74 
6.1 General Remarks . . . . 74 
6.2 Determination of Projection Centre Coordinates . . 77 
6.3 The NRC Approach (Schut) · . . . . . . . . . . . . . . . 78 
6.3.1 Formation of Strips from Independent Models ... 78 
6.3.2 Strip and Block Adjustment (Schut) . . . . . . . . .. 84 
6.4 The Stuttgart Program PAT-M 
6.5 The E.M.R. Program (Blais} . 
6.6 Comparison of the Three Approaches . 
7. Bundle Adjustment . . . . . . . 
7.1 General Remarks 
7.2 USC&GS - Block Analytic Aerotriangulation .. 
8. Conclusions and Outlook .. 
iii 
89 
• 95 
100 
• 108 
• • •• , 08 
• 109 
. 1, 6 
I. Introduction 
1.1 Purpose and Definition of Aerotriangulation 
There are basically two definitions, a classical one and a more 
modern one. 
Classical: Aerotriangulation means the determination of ground control 
coordinates by photogrammetric means thereby reducing the terrestrial 
survey work for photo-control. 
With the advent of electronic computers and sophisticated numerical 
procedures this definition is somewhat enlarged and does not ~apply 
for p~~oviding control for ,photog:rammetric mapping. 
Modern~ Photogrammetric methods of determining the coordinates of points 
covering larger areas. 
This means large point nets. 
1.2 Applications of Aerotriangulation 
1) Provide control for photogrammetric purposes for both small scale 
and large scale maps. 
a) small scale mapping: 
whole countries scales 1:50,000 or so 
required accuracy 1 t 5 m 
b) large scale mapping 1:1,000 · 1:10,000 
required accuracy 0.1 ; 1 m 
2) Point densification for geodetic, surveying and cadastral purposes 
(3rd & 4th order nets) 
3) Satellite photogrammetry; determination of satellite orbits; world 
net ,(H.H. Schmidt) 
1 
2 
1.3 Some Remarks Concerning the History of Aerotriangulation. 
Approximately in 1905 the idea of radial triangulation appeared. 
The first numerical attempts have to be credited to Sebastian 
Finsterwalder at around 1910. After World ·war I spacial aerotriangulation 
was introduced (Nfs:ltri had a patent in 1919). Up until 1935 analogue 
experiments continue (Multiplex!) From then on the idea of base-
change in plotters was utilized for larger projects, mainly in Germany 
(Zeiss) & Holland. 
The real break through for aerotriangulation came with the 
computer (1950's). Some of the key-names associated with aerotriangulation 
are: Gotthardt~ Scht~t·~· Sc:hmidt~;·Brown~ Ackermann. 
Although I shall present to you the graphical mechanical approach 
spanning graphical radial triangulation to Jerie's Analogue Computer 
very briefly, it should be stated now, that these methods are mainly 
historical, although still used. 
1.4 Overview of Methods of Aerotriangulation 
It is obvious, that every step means loss of accuracy. Therefore 
the bundle adjustment with simultaneous solution is theoretically the 
approach which gives the highest accuracy. However, the block adjustment 
with independent models follows closely. 
(') 
Photo 
IMAGE 
H1AG E ---t:> 
FIGURE 1.4 Overview of Aerotriangulation 
Analogue (Plotter} I Digital (Computer) 
rel. orient. ~ Cantilever Extension ! b.- STRIP ADJUSP~ENT ~- ~~~~K AD~. 
MODEL STRIP 1 1 1 wn:n stn ps 
~10DEL 
-
-~---------------- .fi{fJ? 
"ANALOGUE" AEROTRIANGULATION 
' . ----' 
- . -I 
~ 
---
STRIP FQRjltfATION STRIP ADJ. 
SECTIONS 
11 SEMI ANALYTICAL I! AEROTRH\NGULATION 
BLOCK ADJ. (Strips) 
BLOCK ADJ. (Sections) 
BLOCK ADJ. 
(indep. Models) 
--~--~---~~---~~-·~ 
SECTIONS 
STRIP ADJ. BLOCK ADJ. (Strips) IMAGE i t;> MODEL STRIPS 
--·£:"'"BlOCK ADJ. (Sections) 
--~--~""'"" 13LOCK ADJ. 
(Indep. fviodels) 
-t;;:=Bundle BLOCK ADJ. 
111-\Nf\LYTICALli AEROTRIANGULATI 
4 
1.5 Some Reviewing Remarks on Basic Analytical Photogrammetry. 
Analytical Photogrammetry means numerical evaluation of the 
content of photographic images. Being a point by point approach, it 
does not replace analogue photogrammetry but rather complt1ments it. 
Although aerotriangulation is its main application, it is not the 
only one. 
Unlike in analogue work, where it is possible to reproduce the 
physical situation (Porro-Koppe-Principle),the physical situation 
would have to be modelled mathematically. The quality of any numerical 
evaluation depends on how well the mathematical model describes the 
physical situation. 
In the case of photography, the mathematical model is the concept 
of central pnvjection which has the following characteristics: 
projection centre is a point 
light propagates according to geometric optics (straight rays) 
the image plane is perpendicular to the axis of the system 
We all know that these assumptions are not fulfilled. 
- no matter how small the projection centre is (e.g. diaphragm closed), 
it consistsof an infinite number of points. 
- light propagates dually in electromagnetic waves and photons, 
therefore changes in the space to be pf;!netrated (e.g. density changes) 
result in directional changes. (e.g. light passing through glass 
(lenses) changes direction}. 
- the image plane is neither plane nor perpendicular to the system axis. 
5 
. ·. There goes our mathematical model. A better model would mean that 
each camera has to be modelled separately, which would result in a 
special evaluation method for each type of photography - which:is 
ridiculous. 
Therefore we keep the model as it is and adjust our photography 
such that it fits the model. This process is called 11 IMAGE REFINEMENT'! 
Togethere with the basic data for central projection (principal point 
in terms of image coordinates~&:camera constant) the image refinement 
parameters are included with the data of interior orientation. 
These data are: 
- camera constant (linear dependency with radial lens distortion) 
- a newly defined principal point 
- radial lens distortion (symmetric) 
- decentering lens distortion, usually bhoken down into a symmetric 
radial and tangential components 
- film distortion & image plane .deformation 
- refraction 
Contrary to other beliefs, the earth curvature correction is not 
part of image refinement, in fact it is non existent if geocentric 
coordinates are,used. 
The first four data of interior orientation are obtained by 
camera calibration (goniometer measurements or · multi~ll imators etc.) 
in the laboratory. 
Film distortion can best be detected with the aid of a re~eau grid, 
and refraction is a function of flying height, temperature, pressure and 
6 
humidity at the time of photography. The phenomenon of refraction 
has been thoroughly investigated in connection with geodetic astronomy 
and is presented there. 
If, for one reason or a~other some information for the image 
refinement is not available, nothing can be done, but one has to be 
aware that less accurate results are to be expected. 
After the photogrammetric data are prepared to suit our mathematical 
model, computations can proceed. Considering the light rays in vector 
form (P; and TIP;} we come to the most fundamental and most important 
equation, the COLLINEARITY EQUATION. With it any photographic situation 
can be described and it is the base of single photo orientation (often 
called space resection) which may or may not include the parameters of 
interior orientation as unknowns. 
If we want to combine two photographic images to a stereomodel, 
the concept of relative orientation is needed, which mathematically is 
described by a very important condition, the COPLANARITY CONDITION. 
This condition just states, that the two sets of collinearity 
equationsdescribing image rays of the same point and different photographs 
span one plane, with other words, the rays intersect (space intersection). 
The formation of model coordinates is then just the ap.pl'ication 
of the coplanarity condition to all points in question. 
This leads to a model which is similar to the actual situation 
out located somewhere in space. The absolute orientation has the purpose 
to bring it down to earth, which means application of 
7 
- a scale factor 
- a space shift (usually in terms of the three components of the three 
axes of our 3D-space) 
- a space rotation {also in terms. of three rotations around the same 
coordinate axes) 
For aerotriangulation we often have relatively oriented models in 
space usually transformed into one common system, e.g. the system of 
1st model (u•, v•, w• instead of x•, y•, f). 
To form a strip, a scale, transfer is performed by comparison of 
one or more common distances. Having a common projection centre, it 
is theoretically sufficient to compare one elevation only. After the 
strip is formed, it can be absolutely oriented as a unit. 
Of course, it is not necessary to mathematically follow the 
analogue steps of forming models etc. In this case the coplanarity 
condition is not explicitly needed, however the intersection of rays 
is still to be maintained (optimization). Modern solutions using the 
photographic image as a unit perform a total transformation to the 
ground. (simultaneous adj. , bundle approach). 
This again indicates the need for ground control, which can be 
in different coordinate systems. Finally, the need and concept of 
interior orientation remains and cannot be separated from exterior 
orientation unless in the laboratory. 
8 
2. Graphical and Mechanical Methods of Aerotriangulation 
2.1 J3asj __ ~ _ _lgea of Radi~L_Tri ~_gJ:.!l aq on 
Consider vertical photographs, then the principal point of the 
photograph is also the image of the nadir point. A set of angles from 
this point would exactly correspond to actually measured angles on the 
ground. Using several such sets from continuous photographs a triangu-
lation chain can be established with the principal points as radial 
centres and freely chosen points in the overlap area of 3 pictures. 
Practically this is done by transforming the principal point 
of the neighbouring pictures onto the photograph. It is important to 
be quite careful, because the exact identification of the points 
influences directly the accuracy of the result. Then the sets are 
usually drawn onto transparent paper for each photograph and placed such, 
that the directions between principal points coincide. 
9 
/ 
/ 
At the beginning the distance between the first two centres 
can be chosen randomly. However, from then on it is fixed, since 
always the rays of 3 pictures intersect in one point. 
If for the whole strip 2 control points are known, the scale 
of the lay-out and the direction can absolutely be determined, and 
with it we have all the corner points as points in the terrestr5.al 
system. If we have several adjacent strips with side lap, the 
solution becomes quite good, since a fair amount of overdetermination 
occurs. Besides the principal point, also the nadir point or the 
isocentre can be used as radial centres. Usually we do not have exact 
vertical photography. The selection of one of the other points is based 
on the height differences in the terrain and on the tilt of the photography. 
Usually approximations for these points are obtained from the 
photographic picture of a level bubble at the time of exposure. A 
more exact determination (e.g. tilt analysis) is not economical for 
this purpose. 
10 
2.2 Methods of Radial Triangulation 
2.2.1 Graphical Method 
The directions are directly plotted from the aerial photographs 
onto transparent paper. Then the system is laid out onto a base map 
which includes the coordinate axes and the control points. The method 
is quite cumbersome for large nets, especially due to difficulties in 
adjusting the error indicating figures. The accuracy is not very high. 
2.2.2 Numerical Methods of Radial Tri.angulation 
Depending on whether the principal point (or the fiducial 
centre), the nadir point or the isocentre are used, several methods can 
be distinguished. The adjustment is based on the side condition, as 
used in terrestrial triangulation. 
Although the directions can be measured to + lc using a 
radial triangulator .. , e.g. Wild RTl which utilizes angular measurements 
with stereo viewing, the directions have remaining errors of several c 
due to tilt of the photography. It is left to you to determine the 
accuracy. 
2.2.3 Slotted Templets 
This is a purely mechanical adjustment. Using heavy cardboard 
or similar material, the radial centres are punched ~s circular holes, 
the directions as slots. The directions to existing control are also 
cut as slots. On a baseplan, onto which the control points were plotted, 
the templets are laid out and connected with studs. The control points 
stay fixed (pricked or nailed down) while the other studs can move such as 
to minimize the stress in the lay-out. Their fixed position is marked 
through the centre of the stud (pricked). 
11 
0 
n I .. I I 
u 
/ 
Since this is the mechanical equivalent to a least squares 
adjustment, it shows that strips have a rather weak stability (towards 
the side) in the centre unless they are controlled in this area. By 
using parallel strips, the centre becomes strengthened, and less control 
is necessary to support the block. 
r The instrument used for centering is called Radialsecator 
(e.g. RSI- Zeiss which allows scales between 1:2 and 2:1 and tilt 
correction up to 30%). The slots are 50 mm long and 4 mm wide. 
With good material an accuracy of 1% of the horizontal 
distances can be obtained, which is sufficient for rectification. The 
advantage of this method is its simplicity. Limitations are the border 
of the working space. 
2.2.4 Stereo Templets 
A further step leads from slotted templets to stereo templets. 
A stereo templet consists of two slotted templets of the same stereo 
model, which includes the same selected four corner points. However 
opposite corners are chosen for centre points. 
12 
The stereo model has to be relatively oriented (otherwise it 
would not be a stereo model) and approx. absolutely oriented. The scale 
is still free for small changes. The layout of stereo templets follows 
the one of slotted templets. 
Stereo templets can also cover 2-3 models if they are analogue 
and obtained together, e.g. at multiplex. 
2.2.5 Jerie's ITC-Analogue Computer 
The mechanical analogue block adjustment, as developed at the 
ITC in Delft is a further development of the stereo templet method. In 
this case, a photogrammetric block is divided into near square sections. 
For each section double templets are cut, which represent the panimetry 
of four tie points with the other sections and possibly additional points 
within the section. 
13 
The templets are not directly connected but by the use of 
multiplets, which are 4 springloaded buttons on a carrier which can be 
shifted in direction of the coordinate axes. Therefore, the carrier 
can always move to such a 'POSition where the resultant vector of all 
forces is equa 1 to zero. After fixing a doub-1 e ! .. temp let onto the buttons, 
the carrier will move to the adjusted point position. At the same time 
the double templets will shift such that they fit best to the conditions 
(due to resultant forces). In the final phase all sections of the block 
will be at a position which representsthe results of a rigorous adjustment. 
In order to determine the transformation elements required for 
the connection of the sections, Jerie uses the residuals to the preliminary 
orientation with high magnification. Therefore all transformation 
parameters are obtained with magnification and can be graphically obtained. 
Then a new numerical orientation is performed. By repeating the procedure 
with again a magnification the accuracy can be increased without limits. 
Due to uncertaincies in the longitudinal tilt, the same principle 
cannot be applied to vertical adjustment. Therefore a three dimensional 
arrangement has to be used in order to simulate elastic deformations of 
a body in space. At the IGN (Institute Geographique National) in Paris 
such a system utilizing plastic supports has been developed. 
14 
3. Methods of Strip Triangulation and Instrumentation 
/<h 
3.1 The Pr-it:_1_slple of Continuou~--~!n,~ilev_EEr Extension with Scale Transfer 
The idea is to reconstruct the exposure situation. 
lst model: rel. & abs. orientation 
2nd model: 4epen~ent pair relative orientation 
using KIII' byiii 
<PIII' bziii 
WIII 
The problem is bx. This is initially randomly chosen ( ... in figure), 
then the point or points common to models 1 and 2 are set to have the 
same elevation by changing bx. 
Ih_e_r.~fQre __ : Strip triangulation is nothing else but a continuous 
application of cantilever extension (dependent pair relative orientation) 
and scale transfer (base components). 
If there were no error propagation, this would be just perfect 
(e.g. Multiplex). 
Scale transfer is basically the comparison of a distance in 
both models with changing of the ba~e length until the distance is the 
same in both models. 
15 
Any two points in the common overlap area of the two models 
can be chosen. By selecting one rather unique common point (the 
projection centre) only one other point is necessary because the z-
values are directly comparable distances. 
The base is shifted until the elevations are the same. 
tr I . 
I ! l 1 I I 
~7 
I 
l ! 
Since the base extends primarily in x-direction, this means basically a bx-
strip. However, the other base components have to be changed in the same 
ratio. 
Example: Shift 3rd multiplex projector until there is no more x-parallax 
on the measuring table, which has its elevation from the previous model. 
This parallax can be either objective or subjective: 
Objective: Subjective: 
?x 
The eye recognizes the object 
in stereo, but sees two floating 
marks. 
16 
If you use elevations for scale transfer, make sure that the 
counter remains connected and the same! 
If a plane-distance is used, well defined points have to be 
utilized. 
3.2 Triangulation Instruments 
1934 Multiplex was the first triangulation instrument. It is 
however, a low order instrument, as you all know from previous experience. 
"lst order" plotters for aerotriangulation use in effect the same principle 
but with only two projectors. This means base change and image change. 
left right 
CD ® 
base in 
\ 
base out 
change of viewing 
17 
Then 
base in {normal viewing) 
base out (changed viewing) 
Since z is fixed, the same reference height is maintained for scale 
transfer. The problem lies in x/y-coordinates. 
to x: Measure the point in lst model and record value, then disconnect 
x-counter, orient the 2nd model with elements of photo 3, perform 
scale transfer. Then set measuring mark onto point and connect 
x-counter. 
~: There are two opinions: 
a) do not disconnect y-counter 
b) disconnect the y-counter in same manner as the x-counter. 
Both ways are correct, if the base component bx is exactly 
parallel to the x-axis of the plotter. Otherwise secondary errors 
are introduced if opinion a) is taken. It also can be used as 
instrumental check. 
------··-e> )( 
'* geoid (ref. for elevations) 
Ellipsoid (ref. for planimetry) 
18 
for longer strips: 
z ~ H 
therefore, earth curvature is considered an error. It is quite obvious, 
that for a longer strip the b2-range is soon insufficient. 
A similar problem occurs, if the 1st model was not or 
incorrectly absolutely oriented, then the b2 range is also quickly 
insufficient. The b2 limitation is therefore the main reason for an 
absolute orientation of the first model. 
What can be done? ---. b2 shifts 
b2 shift means also change in height reading. 
change height counter after scale transfer (change b2 and zl) 
""~>- Start with a high b2 -val ue 
-"-~Start with an initial <j>-rotation (thendep. pair rel. or.) 
This will cause wrong height readings and lead to projection corrections. 
19 
Another possibility is to break the strip 
~~ has to be measured resp. set in the instrument. 
e.g. Turn a 2g ~-rotation,but the dials are not always that reliable. 
Better compute x-shift using ~ and elevation. Then shift x by 
introducing ~x and turn with ¢ screw until points coincide.The best way is aero-levelling, which means working with a 
fixed and constant z. 
~~ is the convergence due to earth curvature (~ 1 c/km base length) 
Similar considerations might be necessary for the by range 
(e.g. in'itial K-setting or by-shift!) 
20 
3.3 Strip Triangulation with Plotters Without Ba~e:chaoge 
: 
~·· 
Since .the rotational 
centre is fixed, only a 
H movement is possible. 
Problem: The absolute orientation of the right projector has to be 
reconstructed in the left projector. 
This would be not much of a problem if there were base 
components. 
Just use a cross-level, and in this case level plate with 
~ and ~ or (wl and WR) 
Since the rotational centre is a fixed instrumental point, a 8Z0 -value 
has to be introduced to compensate the parallel shift. 
This is an iteration, since b2 will only be obtained after scale transfer, 
when using Z. If the scale transfer utilized a planimetric distance, then 
8Z0 can be directly computed. 
Further to this cumbersome approach: Additional instrumental 
errors have to be considered (both projectors are not exactly the same). 
Since the A-8 does not have base components the instrumental 
base cannot be rotated but is a straight line along the strip. Therefore 
if the flight line looks like this: 
21 
system by a series of Helmert transformations. 
Following this, it is better to use ipdependent models 
and completely compute the strips, even in elevation. 
3.4 Precision Comparators 
Jena 1818 Stereo Comparator 
Accuracy: + 10 em for .X' andy' 
~ 3 em for px, py 
(several observations) 
Working sequence: 
1) placing and clam_iping of photo plates 
2) fix eye base obtain a parallax free image of the floating marks 
in the measuring plane 
3) Make fiducial lines parallel to instrument axes (k -rotation) 
22 
4) Read or set the numerical values for fiducial centre (representing 
principal pt.) 
5) Drive to image point in left photo, using x• andy' motions 
6) Obtain stereo coverage by moving right photo with Px and Py 
7) Read or register,:'X', y•, px, py. 
23 
Although comparators are very easy in principle and use, they 
demand a great mechanical effort. The following conditions have to be 
fu1fi 11 ed. 
1) Straight and easy movement of all carriers 
2) Parallelity of axes for left and right photos 
3} Perpendicularity of x and y axes 
4) Image plane has to be parallel to axes 
5) Parallelity of lead screws and measuring spindles to image motions 
6) Precision division of scales and of spindles 
Comparators are calibrated with the use of grid plates, 
which have to be more accurate! 
Most stereo comparators follow the same construction principle 
with higher magnification etc. such as 
~lil d STK-1 
OMI comparator 
SOM comparator 
Hilger and Watts comparator 
The Zeiss Oberkochen PSK has newer construction principles and is much 
more compact 
The photoplates (1) are clamped unto precision glass-grid 
plates (2) which are vertical. 
The grid enables the coordination of image points in units 
of its grid, which is 5 mm. 
24 
x. 
Abb. 10. Schema der linken Hiilfte des Priir.isions-Stereokornparat<>rs von Zeiss (Werk-
zeickntt11{1 Carl Zei88, Oberkochen). 
(from: Schwi defsky "Photogrammetri e") 
Since both measuring diapositive and grid plate are made 
from glass, temperature changes do not effect the accuracy (illumination!!) 
The fine measurement is done with the aid of a fine scale (4) onto which 
both photo and grid is imaged through the optical system (3). In the 
same optical plane the floating mark (better measuring mark, since the 
comparator can be used mono and stereo) is positioned. 
Two measuring spindles via level;-s (6)and (7) can move the 
measuring scale in x andy directions until grid line and measuring 
scale march coincide. Large gear transmissions permit reading to 1 ~m, 
which is mechanically possible since the maximal way is 5 ~m. 
8 to 16 x magnification is possible. 
25 
There are also a series of mono comparators, most of which again 
fol1ow the basic carrier prindple as discussed. 
Just a few words to the 
NRC - mono Comparator 
(built by Space-Optics and sold by Wild) 
The Model 102 Monocomparator accepts either glass plate diapositives 
or film in up to 911 x 911 format. 
Measurement of image points in X and Y coordinates are made with 
respect to precision scribed measuring marks on a glass plate. The 
measuring marks are spaced at 20 mm intervals in a 12i1' x l2'11·'i matrix. 
The photo is positioned with respect to a specific measuring mark location 
by slewing the grid plate and photo on an air supported carriage. The 
precision coordinates are measured with respect to the referenced 
measuring mark by two short precision lead screws with 20 mm of travel. 
A measurement consists of a macro movement of the grid plate and a micro 
adjustment of the lead screw. Air supply to the carriage is controlled 
by a foot pedal. The lead screws are moved by finger touch measuring 
disks. The X and Y coordinates are continuously displayed and are recorded 
on any digital storage device such as punched card or paper tape. 
Identification numbers may be inserted along with the X":and Y coordinates. 
26 
The design concept of the Model 102 Monocomparator uniquely 
combines high accuracy with high productivity. 
Those factors which contribute to accuracy are: 
Short lead screws (20 m) 
Precision glass measuring mark plate 
Selected materials with uniform thermal coefficient of expansion 
and high thermal conductivity 
Adherence to Abbe•s principle 
The Monocomparator has an overall accuracy of~ 2.5 micro-
metres and a repeatability of measurement in the order of+ micrometre. 
27 
A special construction is the DBA- Monocomparator which 
utilizes the measurement of distances. 
(Multilaterative Comparator) 
By changing the plate into all 4 possible positions. the point is 
determined by 4 dista.oces. The overdetermination permits self-
calibration (det. of comparator parameters by LS-adjustment). A 
program for obtaining photo coordinates (and necessary camp. parameters), 
is supplied by DBA. 
Theory of Operation {according to DBA) 
The theoretical basis for the comparator is best illustrated 
with the aid of Fig. 1 which shows the four measurements rlj' r 2j' 
r3j' r4j of a point xj' yj. It will be noted that in ,the actual 
measuring process, the pivot of the measuring arm remains stationary 
while the plate is measured in four different positions. This is 
precisely geometrically equivalent to a process in which the plate itself 
remains stationary while the pivot assumes four different positions. 
The actual measurements rij are from the zero mark of the scale to the 
point rather than from the pivot to the point. To convert the measurements 
to radial distances from pivots, one must specify the radial and tangential 
offsets (a, s) of the pivot relative to the zero mark. c c If x1, yi denote 
the coordinates of the pivot corresponding to position i of the plate 
28 
(i = 1, 2, 3, 4), one can write the following four observational 
equations relating the measured values rij and the desired coordinates 
xj' Y j: 
(rlj + a)2 + 13'2 = (xj - X~)2 + (yj - y~)2 
(r2j + a)2 + ~,2 = (xj - x~)2 + (yj - y~)2 
2 2 c)2 ( c)2 (r3j + a) + ~) = (xj - x3 + Yj - Y4 
(r4j + a)2 + ~2 = (xj - x~)2 + (yj - y~)2 
These equations recognize that since there is in reality only one pivot 
and measuring arm, a common a and1j apply to all four positions of the 
plate. If the ten parameters of the comparator (i.e. a and!!~! plus four 
sets of x:, y: were exactly known, we could regard the above system as 
1 1 
involving four equations in the two unknowns xj' yj. Accordingly,the 
process of coordinate determination in this case owuld reduce to a 
straightforward, four station, two-dimensional least squares trilateration 
In practice, the parameters of the comparator are not known to 
sufficient accu~acy to warrant their enforcement. It follows that 
they must be determined as part of the overall reduction. This becomes 
possible if one resorts to a solution that recovers the parameters of the 
comparator while simultaneously executing the trilateration of all 
measured points. Inasmuch as one is free to enforce any set of parameters 
that is sufficient to define;·uniquely the coordinate system being employed 
three of the eight coordinates.of the pivots can be eliminated through 
the exercise of this prerogative. In Figure 1 we have elected to define 
they axis as the line passing through pivots 1 and 3 thereby making 
29 
x~ = x~ = 0. Similarly, the x axis (and hence also the origin) is 
established by the particular line perpendicular to the· y axis that 
renders the y coordinates of pivots 2 and 4 of equal magnitude but 
opposite sign (thus y~ =. -y~}. This choice of coordinate system has 
the merit of placing the origin near the center of the plate. 
By virtue of the definition of the coordinate system, only 
seven independent parameters of the comparator need be recovered. · If 
n distinct points are measured on the plate, the above equations may 
be considered to constitute a system of 4n equations involving as 
unknowns the seven parameters of the comparator plus the 2n coordinates 
of the measured points .. When n > 4, there will exist more observational 
equations than unknowns, and a least squares adjustment leading to a 
2n + 7 by 2n + 7 system of normal equations ~an be performed. Although 
the size of the normal equations increases 1 inearly \'lith the number n 
of measured points, their solution presents no difficulties, even on 
a small computer. This is because they possess a patterned coefficient 
matrix that can be exploited to collapse the system to one of order 
7 x 7, involving only the parameters of the comparator. Once these have 
been determined, as independent, four station, least squares trilateration · 
can be performed to establish the coordinates of each point. Details of 
the data reduction are to be found in the reference cited below.* 
* Reference: D. Brown, 11 Computational Tradeoffs in the Design of a 
One Micron Plate Comparator'', presented to 1967 Semi-Annual Convention 
of American Society of Photogrammetry, St. Louis, r~o. , Oct. 2-5-. 
Available upon request from DBA systems, Inc~ 
30 
Suffice it to say here, that the internal contradiction resulting from 
the redundancy of 1ttle~1neasuring process can be exploited to effect an 
accurate calibration of the parameters of the comparator for the 
particular plate being measured. Hence, the designation of the instrument 
as a self~calibrating multilaterative comparator. 
The Computer Program 
A fortran source program is provided with the comparator 
to reduce raw measurements to comparator coordinates. The program is 
designed so that a minimum of modification is required to adapt it 
to almost any computer. One version is designed specifically to run 
on a minimal computer configuration such as an IBM 1130 with card 
input. Another version is designed for medium to large scale computers. 
Both feature automatic editing and rigorous error propagation. Typical 
running time on an IBM 360/50 for the reduction of a plate containing 
25 measured images is well under 30 seconds. 
In addition to producing the final coordinates and standard 
deviations of the measured points, the program generates.the four 
measuring residuals for each point and the rms closure of trilateration. 
The residuals provide a truly meaningful indication of total measuring 
accuracy, and the rms error of the residuals, representing as it does 
a rms error of closure, provides a particularly suitable criterion for 
quality control. 
31 
y 
n 
Figure 1: Jllustrating Geometrical Equivalent of Multilatera~ivJ 
·Comparator. 
3.5 Point Selection, Transfer, Marki~g, Targetting etc. 
The adequate preparation of the available photography for 
aerotriangulation purposes is very important. The prime requirement 
for aerotriangulation is that the photographs overlap. Usually a 
60% overlap is available (often 90% in flight and then every 2nd 
photo is discarded). The overlap between strips is 20 - 30%, sometimes 
60%. 
1) Layout 
Contact prints of the whole strip (or block) are laid out 
and examined for overlap areas {thinning of 90% overlap, gaps?) 
deformations (large ones will show), general information, e.g. scale, 
topography etc. 
32 
2) Selection of triangulatron·pprnts in the·overlap areas 
< ' a: ' 
- pass points = common points in three or more consecutive photographs 
within the strip (pass from model to model) 
- tie points = points located in common overlap area between strips. 
Usually the tie points are also pass points. This however, 
requires a regular strip and block pattern. 
The pass points are selected such that they are located in 
the vicinity of the V. Grubei~'Jrb.twts:,, used commonly for relative orien-
tation. This means 6 points per model. 
If the strip is measured in stereo (e.g. analogue triang., indep. 
models on plotter, stereo comparator), then only three points per 
photo have to be marked, whereas·· for mono comparator measurements, 
Bine points per photo are required (except for lst and last in strip, 
which req. 6} 
f , ... ~. 
:O 
~ ® 
0 0 
Stereo 
33 
The tie points, which are often passpoints, should be selected in 
the centre of the common overlap area (relief displacement etc!) 
Since the stereo models are not created across the strips 
it is necessary: to mark tie points in· bOth strips! . 
More than 30% side lap strengthens the vertical behaviour 
of a block, and with 60% the principal points are included as tie 
points also. 
The points selected are usually natural terrain points, 
because they provide the highest accuracy. However, the following 
has to be kept~~n mind: 
-do not choose a point on top of houses, trees etc., if not absolutely 
necessary. It is easier to: 
- measure ground points, such as 
road and/or railroad intersections or junctions. dunctions of ditches 
or other characteristic line features~ Detail points, e.g. rocks, 
small bushes, high contrasts edges or corners of shadows (again high 
light/dark contrast in photography +be aware of time lapse between 
strips!) 
3) Marking of points 
a) ~1anua 1 
- Circle the area on the contact print with easily recognizable marker 
- Provide a sketch of the-details of the area surrounding the selected 
point (often on back of so-called "control" print) 
- describe point, if necessary (e.g. 11 top step 11 ) 
34 
' 
' J~nction of Ditches 
South of Road 
Near old Barn 
It is important for the operator, that all pass and tie points are 
marked as well as existing ground control. The latter is often 
specially coded according to its nature (e.g. hor. or vertical or 
both). 
b) Mechani ca 1 
There are several instruments of point marking, e.g. 
Wild PUG 
Zeiss (Ober~ochen} Snap Marker 
Kern PMGl 
Zeiss (Jena) TRANSMARK 
They range f,rom quite simple and punching a hole into the 
emulsion to quite sophisticated and burning a hole using laser light. 
Stereo markers such as the PUG have to be used in order 
to correctly identify points for mono.-comparator work. In this case 
the hole is drilled. 
c) Targetting (Pre-signalization) 
For highest accuracy, all photogrammetric triangulation 
points should be targetted on the ground. This eliminates the error 
contributed by point transfer which can be several~m. Targetting 
35 
is usually done for those points for wh'ich coordinates are desired 
(e.g. control densification by photogrammetric means,cadastral 
surveys, construction projects,urban mapping) whereas pass· and 
tie points are usually not pretargetted. Pretargetting requires 
field work and has to be done only a few days before photography in 
order to keep the loss of targets down (people interference). The 
important thing is to create a ~ood light/dark contrast. 
The simplest way is using paint (crosses on highways, 
or direct painting of survey menumell'l't~J. Card board, plastic or 
fabric targets are also in use. 
Examples for targets 
36 
It is quite obvious, that the target size has to match the scale of 
photography (resolution, identif.) Target should be 50- 100 vm 
on photo~raph with the central part being at least 25 - 30 vm in 
diameter. 
Except when there are not many characteristic surface 
features, pretargetting is not used for topographic mapping. 
Sometimes a lower reconnaissance flight and 35 mm film is 
used to obtain detail pictures of the surroundings of points. These 
pictures are then used to identify the points in the small scale 
compilation photography. (Correlation with Zoon- Transfer Scope 
[Bausch and Lomb]). 
37 
4. Error Theory of Strip Triangulation 
4.1 General Assomptions 
Error theory is always the base of adjustment. Although 
each model is deformed due to errors in interior and exterior 
ori en tat ion this simplified theory propagates that a 11 errors are 
caused by transfer from model to model. The assumption is justified 
because the actual model deformations for aerial photogrammetry are 
in the magnitude of 1 - 10% of the transfer errors as empirically 
determined. 
This means basically, that the 7 elements of transfer are 
associated with errors, namely: 
Scale transfer error ilS. (i = 1 ' 2 ... (n - l ) ) 1 
Azimuth - transfer error M. 
1 
because of (n-1) 
Long itud ina l transfer error M· connection for 1 
lateral tilt transfer error llW. n - models 
1 
x-shift error tJ.X. 
1 
y·~shift error ll y. 1 
z-shift error £l Z; 
4.2 Vermeir•s Simplified Theory of Transfer Errors 
Vermeir (ITC) considers only the errors in ils, ~¢, ilw 
ila and neglects the shift errors, since they are small. The strip 
axis is used! Consider~ng azimuth errors, the deformation of the 
strip axis is as follows: 
38 
numerically: 
!sA ··(!)·. = 6A (j) = 6A (j) 
6A (!) = 6A (f) + 6a1 = 6A (j)+ 6a1 
6A Q) = 6A ® + 6a2 = &A Q)+ &a1 + 6a2 
- - - - - - - - - - - - - - -
6A i = 6ACi:v + 6ai-l = 6A(D+ 6a1 + Lla2 + ... + 6ai-l 
therefore: 
i=l 
Similarly for ~- tilt: 
1 11l 1 (f!t:) 
! 
\Y.-:: r;.. ~"!.. 
-~:...:.._---It-_ ~ ~® 
--· ··-- ~·lr't-··--··---···· .. ·····'··········-······ .. ··-.. ····· .. ·····•··········· ·············-·····---·· ····-··+-···--·-···-······················1-···---··-· .. ····------.~>- x 
A ~ ~Af, 2 3 4 5 
<D 
this leads to: 
and also for~- tilt: 
Scale transfer errors: 
t 
I L _____ -i- - - - ·-
39 
I 
I 
I I 
- --=-- -_:' _j 
therefore 
i -1 
[1$ CD = b.S Q) + \l~l 
40 
ilS \) 
4.3 Coordtnate Errors in Stri~ AXis 
"" . < 
Now we can consider coordinate errors: 
= ily 
0 
L\y 1 = L1y 0 + b CD ilA G)= L1y 0 + b Q) ilA <D I 
i.. I 
t1y2 = l1Y1 + ~ ([JilA @= t1y0 + b (J)l1A(D+1 b(g~~ 
_ _ _ _ _ _ _ _ _ _ _ _ _ l _ ~A(!)+ ila1 
t1y1 = L\y1 _1 + b(DM{n= L1y0 + b Q)LlACD+ I b@l'l'@ + ••. + 'CD Ll'tD 
. . . initial errors .I transfer errors 
i l 
l'lY; = l1Y0 + ~~l b@ ilA@ 
= l'l y 
0 
b ::: constant! 
i 
r 
Jl=l 
41 
i ]l-1 
tJ.y 1• = !:iY + X. tiA .(j)+ b .. L: r 
.. 0 1 .... 
- v=l .v"'l 
tJ.a . 
\1 
This is the famous double summation of random errors, which 
leads to a systematic behaviour, as will be shown later. 
There is another way of representing the same error: 
Since this formula represents directly the same as the previous one, 
the double summation characteristics are still there, only hidden! 
Similarly: 
i 
/:iX. = !J.X + l: 
1 0 }.1=1 
= l:ix + x. 0 1 
i }l-1 
bQStJ.S® 
I 
l:iSCD+1 b .l: 2: }.1=1 ·v==l 
I i -1 
= tJ.x0 + .x,. !:iS CD +1 r •· (.x. - x ,1 v= 1 , '> J tJ.s ~ 
and 
i 
= 6Zb + .. L: 
u=l 
42 
= ~z. + .x. 0 1 
i p-1 
Mrp+l b L: .!: ~<jl 
\.!) p=l '1/=1 )) I i _, 
= 1../!,0 + X; M.;(i) +I· L (.x ... X. )M 
v=l 1 v .v 
initial deformation 
error due to transfer error 
Before covering the off axis errors, I would like to say 
a few words about double summation. 
4.4 Double Summation 
If one considers, for example the azimuth error ~a as a 
random error Ea, then their accumulated effect in'·the llth model 
wi 11 be: 
E = E l + E 2 + ... + E . ap a a a 1 }.1-
}.l-1 
= L: 
:v'=l 
These single errors Ea. in turn, Gr~qte a lateral error. 
1 
. . . + E .) = b Y1 
i 
.L: 
p=l 
E . 
Yll 
\4e hav~ ~rrors which are obtained by double summation of a series of 
random errors. 
i 
a. = .L·· 1 
v=l · 
Generally: 
v~l · 
L: €. = ( i -1 :~ + ( i - 2 J 
1 
) 
43 
and this series has certain systematic characteristics. This was 
first discovered by Gotthardt (1944- rolling dice) and Roelofs (1949 
drawing lots) with the aid of statistical experiments. Moritz (1960) 
gives a theoretical explanation based on the fact, that all xi are 
strongly correlated by the s;· 
Here are three pract"ical examples taken from Finsterwalder 
Hoffmann Photograrnmetrie, p. 370. 
1
_' b~i~=-;;~~~~~-EI~~-
1 + 3 -+- 3 + 3 _ !) - 9 __ 9 --- 7 1 --7 -- 7 
2 + 1 + 4 + 7 - 6 ---15 -- 24 + 4 ---3 --10 
3 -12 - 8 1 + 8 - 7 - 31 + 8 ·Hi - 5 
4 + s: - 5 6 - 2 - n -- 40 + 2 +7 + 2 
5 + 3 -- 2 8 -j-15 -1- 6 -- 34 ·- 3 +4 + 6 
6 + 3 + 1 - 7 + 3 + n - 2;:; -- 9 -- .'j + 1 
7 + 4 + 5 --· 2 -- 3 + 6 19 + 1 --4 -- 3 
8 ·-· 7 --· 2 4 + 6 + 12 -· 7 + 5 + 1 ·- 2 
9 -- 7 -- 9 -- t3 +11 +23 + 16 + a -H + 2 
10 - 8 ---17 -- 30 . 0 +23 + 39 -- 4 0 + 2 
1l 0 - 17 --- 4 7 - 6 + 17 + 56 - 2 -·- 2 0 
12 -- 4 --21 - 68 + 3 +-20 + 76 + 6 ·1-4 + 4 
13 + 1 --20 - 88 +10 +30 +106 -·- 9 --5 -- 1 
14 + 5 ---15 ---10:3 - 2 +28 +134 + 8 --2 --- 3 
15 0 -15 --118 --12 +16 +150 +10 ·f-8 + 5 
16 ... 5 --20 --1:m ---10 + 6 +156 - 7 -t-1 + 6 
17 + 5 --15 ---Hi3 ·- 8 ·-· 2 +154 --10 ---9 -- 3 
18 + 3 --12 -165 +11 + 9 +163 + 5 --4 --- 7 
19 -- 9 --21 --186 -- 2 + 7 -j-170 +10 +6 
20-1 -22 --208 -7 0 +170 --5 +1 0 
H --· 1 --23 ---231 -· 3 -- 3 + 167 
22 ·- 3 --26 . --257 --- 2 -· 5 + 162 
23 -- 7 ---33 ---2!)0 + 2 -- :{ -t-15!) 
24 . - 9 --42 ---332 - 7 ---10 ·t-149 2~--=---=~~--- --=~~=-------~~---~~-~-+ 146__j _________________ __ 
44 
It is evident, that the two error series are random, but 
show systematic character after double summation. 
This is a statistical statement, as can be shown with a 3rd 
series, which remains random! 
4.5 Off Axis Points and.Their Coordinate Errors 
a} Scale Error: 
Errors in y and z (the x-error has already been considered 
for the nadir points) 
i-1 
t:.y. = y. ~:.~= y.{t:.SJl + E t:.s ) 
1 1 (!) 1 Ur v= 1 v 
i -1 
t:.z • = z . 6~ = z . ( t:.S:p + E t:.s ) 
1 1 \!) 1 \.!.-I v= 1 v 
" . 
b) Azimuth error: 
i-1 
t:.fv.:\ = -y. (t:.A"' + E (!) 1 - \.!) \.) = 1 
t:.z. = 0 . , 
c) Longitudinal error: 
t:.x. 
1 
l:!y. 
1 
i·l 
= Zi M>Q) = Zi (6~f.j\ + E 
I..!) v= 1 
= 0 .. 
t:.a ) 
'\) 
8¢1 ) 
\.) 
45 
d) Lateral error: 
i -1 
t.y1• = -z. 6fq::;. = -z. (Aq,'\ + r Aw ) 1 (!_,1 l -zv v= 1 \) 
i-l 
AZ • = y • M4--f1 = y • ( Mf~ + E Aw ) 
1 1 Q) 1 H I -1 V 
....... v-
All these influences combined give the following errors in x, y, z 
for a point in model i, when the origin of the coordinate system is 
close to the 1stprojection centre and the x-axis follows the direction 
of flight: 
I · , · 1 • ., · ,_ 1- ,~ I 
AX= AX0 + XAS;;- YAAj;+ ZA~l + E (x-x )AS - y :E Aa + Z E A$ + r ~ ~) ~} I _,. \) \) -1 \) -1 \) X v- v- v-
'1 ., - ., I 
,_ 1- l-
AY= AY0 + YAi.1D' + XAJlr,)- ZM~f'+ yEAS + E (x-x )Aa - z E Aw + r 
· \.!..- \!J 1 v=l "' v=l "' "' v=l "' J y 
AZ = AZ + 
0 
initial errors 
Now we simplify again: 
= AS = 2 
Therefore 
i -1 
i-1 i-1 i-1 
z r A s r (x-x )A~ + y r Aw + r 
v=l "' v=l "' "' v=l "' z 
transfer errors 
= AS 
= Aa 
pointing 
error during 
measurement 
E (x-x.) As"' for a constant As becomes: 
v=l 1 
46 
i _, 
~ (x-xi)6sv = (x-x1}6s1 + (x-x2)6s2 + ...... + (x-xi_1)6si-l 
:v=l 
= [(i-l)x- (x1 + x2 + ..... + xi_1)]6s 
and if we consider b~ ~ b: 
x.(i-1) 
= [(i-l)x- 12 ]6s = 
x. 
(i-1) (x- r) 6s 
Now, if we just consider the nadir points, then x ~ x. 
1 
i -1 . 1 ~ (x-x.) 65 = l:_x 65 
-1 1 2 v-
x. 
However: i ~ ~ ~ ~ (i = number of models) and for longer strips we can 
set (i-1) ~ i, therefore: 
i-l 6S 2 ~ (x-x;) 6s = 2b x = 6x (major influence!) 
v=l 
This means, that if we have a constant scale error, the influence is 
in form of a parabola. 
The single summation will give: 
i -1 
~ 6av = 6a1 + 6a2 + ........ + Aai-l 
v=l 
with 6a1 = 6a2 = .... = 6a 
47 
= ( i -1) L\a 
z L\a x 
b 
The single summation results in a linear behaviour. Therefore: 
LlS 2 M M_ 
LlX = LlXO + xilrj)- yMID + Zll't1) + 2b X - b xy + b xz + r X 
L\a 2 ils L\w 
L\y = L\y0 + YL'i\1) +XL\~- ZMtj)+ 2b X + b xy - b xz + ry 
L\Z = L\z 0 + zil~f)- XL\~)+ yl\~-* i - ~w xy + L\~ xz + r z. 
If we want to establish a general equation, we can set: 
L\X = ao L\yo = bl\ L\z0 = co 
__ o d 
L\~:~1) = al ill}j) = bl Mj = cl Ll~D = dl 
L\S M_ b2 Llql = 2b = a2 2b - 12b c2 
And now we have: 
L\x = a0 + a1x - b1y + c1z + a2x2 - 2b2xy + 2c2xz 
2 L\y = b0 + b1x + a1y - d1z + b2x + 2a2xy - 2d2xz 
2 ilz = c0 - c1x + d1y + a1z_- c2x 2d2xy + 2a2xz 
L\w - d2 2b -
These functions are not independent, except for the axis where y = z = 0. 
However, we have now some equations which are independent of the models. 
Before, the strip deformations were functions of x, y, z and i!! 
For flat terrain, z =canst.: 
48 
2 ~Y = b0 + b1x + a1y + b2x + 2a2xy 
~z = c0 + c1 + d1y + c2x2 + :d2xy 
Now we make them all plus and omit the 2 since the general coefficients 
are different. 
112 
Transfer errors 1st summation 
~s, ~a, ~cp, ~w ---~ ~S. ~A. M. ~r~. 1 1 1 1 
for b-t~>O J 
Now we can reverse the whole procedure 
1st differentiation 
~x, ~y, ~z 
We had: 
2nd summation 
~x, ~Y, !J.,z 
JI 
2nd derivative 
49 
!J.X = a0 + a1x + a2x2 + - y(bl + b2x + ... ) + z(c1 + 2c2x + .. ) 
!J.y = b0 + b1x + b i + 2 + y(al + 2a2x + ... ) - z(d1 + 2d2x + .. ) 
!J.Z = co + clx + 2 c2x + + y(dl + (2)d2x + .. ) + z(a1 + 2a2x + .. ) 
dtJ.y . dtJ.z . 
!J. !J. _ y( aXlS) + z( aXlS) 
x = xaxis dx dx 
dtJ.x . 
tJ.y = tJ.y + y( ax1s) z(tJ.n ) 
axis dx - (x) 
dtJ.x . 
A A + (An ) + z( ax1s) 
uZ = uZ . Y o~•( ) dx aXlS X 
Compare this with original coefficients: 
!J.X = !J.XA - ytJ.A(x) + Z!J.~(x) 
-
tJ.y = tJ.yA + ytJ.S(x) - ztJ.n(x) 
~ WIIIMIU11IIIIIII!Iillijt 
= 0 for z =canst.! 
(up to 10% of flying height). 
These are independent polynomials, the others are derivatives. 
So far we have only talked about random errors. There are 
also systematic ones, caused by instrument errors, distortions, etc. 
as well as by earth curvature, refraction, etc. 
Historically systematic errors were approximated by a 2nd order 
polynomial. Since the strip deformations appeared to have similar 
behaviour, it was assumed that the main error sources were systematic. 
This lead to intensiv e instrument checks. Today it is obvious that 
systematic errors and random errors with systematic characteristics due 
to double summation are superimposed. The initial doubting came, when 
50 
triangulation with the C8 and radial triangulation showed similar 
behaviour, although they are the result of extremely different procedures 
and instruments. 
Before going to actual strip adjustments, a few words on the 
magnitude of the errors: 
while 
cr~x 
cr ~s '\ 
I 
I 
\ 
r j 
cr~s ~ 0.1 - 0.2 %0 
cr~a = cr~t = cr~w = lc (same cl~im 0.5c) 
increase proportional with /x3 
increase proportional with /x 
According to Ackermann: 
where 
- = ~. 
X b' Y- = 'j_, b' 
- z ( z = b normed quantities) 
There are strong correlations between x. y., and z. as well as with other 
1 ' 1 1 
points. In addition a measuring error of ~ 10 vm in the image scale has 
to be considered. 
51 
5. Strip Adjustment with Polynomials 
5.1 General Remarks 
Orig·inally, strip adjustment was a non linear interpolation 
procedure. This depends strongly on the number. and distribution of 
control points. The problem is, to approximate the real errors as good 
as possible. 
Let us assume, we know the deformation: 
A 
I 
! 
One tries to determine the plausible deformation, using these known 
points. The deformation is considered to be steady. Such a curve can 
for example be obtained using a plastic ruler which gives you a better 
curve than any numerically determined one. However, there is the 
additional problem of measuring errors in the known points. 
52 
'"'"'''""'"""""'"" 
What is better? 
One thing is quite obvious that points close together might 
cause large errors. 
/j 
53 
_ ............ , ..... "'""""""""''"'""""'""'"""-...................... - .............................. ---................................. 1--•·---.. -· .... ·----... -·ild-~ 
f.- loh~er --·~ 
!"; rZ)c ,.,~ 
Extreme case: one model! 
There has to be a certain restriction to the degree of the polynomial. 
54 
5.2 The U.S. Coast & Geodetic Survev Method, as Example 
Extract from: Aerotriangulation Strip Adjustment 
by M. Keller and G.C. Tewinkel 
U:S. Coast .and Geodetic Survey 
AEROTRIANGULATION is a photogrammetric technique for 
deriving ground coordinates of objects from a set of overlapping aerial 
photographs that show images both of those objects and also of a 
relatively sparse distribution of other objects whose coordinates are 
known from previous classical measurements on the ground. Provisional 
photogrammetric coordinates of objects can be determined by at least 
two general methods: 
(1) through the use of a high-order photogrammetric plotting instrument, 
or 
{2) through analytic computations based on observed coordinates of 
images on the photographs. 
The discussion pertains specifically to both cases. In each instance, 
the photogrammetric strip coordinates of points comprise a thick, 
three-dimensional ribbon in space generally not referred specifically 
to any ground surveyed system of points. 
Finally, the strip coordinates, in order for them to be useful, 
must be related to the ground system through the application of poly-
nomial transformations in a curve-fitting procedure to adjust the 
photogrammetric strip coordinates to agree with known ground surveyed 
coordinates. This fitting technique is called Strip Adjustment. The 
polynomials are nonlinear because of the systematic .accumulation of errors 
throughout the strip. The application of least squares provides a 
logical analysis of redundant data. 
55 
This paper comprises a documented computer program concerned 
only with the transformation and adjustment of the strip coordinates 
of points to fit ground control data. 
Technical Bulletins No. 11 and No. 102 presented the principal 
formulation still being applied in the Coast and Geodetic Survey, and 
No. 21 3 included an application ofthose ideas. The present bulletin 
includes helpful modifications which have been added since the dates 
of the original releases, combines the ideas of the three former 
bulletins into a single operation, adapts the program for either 
instrumental or analytica aerotriangulation, and embodies a systematic 
technique for correcting horizontal coordinates for the local inclina-
tions of the strip. 
This program is considered to be the first of a series of two 
or three new programs for analytic aerotriangulation. Chronologically, 
this program will be used as the third step in the provisional adjustment 
of strips. The other two programs will consist of 
(1) the reduction of observed image coordaintes and 
(2) relative orientation, including the assembly of the oriented data. 
The three programs will comprise a complete practical set for analytic 
strip aerotriangulation for use on a medium size computer. 
1 Aerotriangulation adjustment of instrument data by computational methods 
by W.O. Harris, Technical Bulletin No. 1, Coast and Geodetic Survey, 
January 1958. 
2vertical adjustment of instrument aerotriangulation by computational 
methods by W.O. Harris, Technical Bulletin No. 10, Coast and Geodetic 
Survey, September 1959. 
3 Analytic aerotriangulation by W.O. Harris, G.C. Tewinkel and C.A. 
Whitten, Technical Bulletin No. 21, Coast and Geodetic Survey, Corrected 
July 1963. 
56 
INTRODUCTION 
The adjustment of aerotriangulated strips has been the 
subject of numerous articles and publications for a few decades, as 
indicated presently by the references. The articles agree in general 
that the equations for transforming photogrammetric coordinates into 
ground coordinates can be expressed by polynomials which need to be at 
least second degree. Schut4 and Mikhail 5 show that a conformal trans-
formation in three dimensions is not possible if the degree is greater 
than one, although a conformal transformation in a plane is not limited 
as to degree. Inasmuch as an ideal solution of the three-dimensional 
problem does not seem to exist, photogrammetrists have been free to 
devise approximate, quasi-ideal, and impirical solutions that seem to 
give practical and usable solutions to their problems. Several examples 
are cited. 
The Coast Survey formulas applied herein are: 
x• = x - ~z(3hx2 + 2ix + j ) + ax3 + bx2 + ex - 2dxy - ey + f 
y• = y - ~z{kx2 + ~x + m) + 3ax2y + 2bxy + cy + dx2 + ex + g 
z• = z[l + {3hx2 + 2ix + j)2 + {kx2 + ~m + m2]112 + hx3 + ix2 + 
. k 2 JX + x y + my + n . 
(The x, y coordinates refer to the axis-of-flight system after the 
application of Equations 22 and 23). Schut4 states the following 
conformal relations for horizontal coordinates to third degree: 
4 Development of programs for strip and block adjustment by the 
National Research Council of Canada by G.H. Schut, Photogrammetric 
Engineering, Vol. 30, No. 2, page 284, 1964. 
5 Simultaneous three-dimensional transformation of higher degrees by 
E.M. Mikhail, Photogrammetric Engineering, Vol. 30, No. 4, page 588, 
1964. 
57 
. 2 2 3 2 2 3 
x• = x + a1 + a3x - a4y + a5(x -y ) + 2a6xy + a7(x -3xy )-a8(3x y-y ) + 
y• = y + a2 + a4x + a3y + a6(x2-y2)+2a5xy + a7(3x2y-y3) + a8 (x3-3xy~~)+ 
Webb and Perry6 used 
x• = x + a3x2 + b3x + c3xy + d3xz + e3 
y• = Y + a4x2 + b4xy + c4y + d4 
z• = z + a5x2 + b5xy + c5xy + d5y + e5x + f 5 . 
The following equations in a plane appear in Schwidefsky•s textbook: 
y• = y + a2x3 + b2x2 + 3a1x2y + 2b1xy . 
Arthur8 of the Ordnance Survey of Britain published: 
x• = x + a1 + a4x + a6z - a7y + l/2a8x2 + a10xz - a11 xy 
2 y• = Y + a2 + a4y + a5z + a7x + a8xy + a9xz + l/2a11 x 
z• = z + a3 + a4z - a5y - a6x + a8xy - l/2y10x 2 
(3) 
(4) 
(5) 
Norwicki and Born9 suggest variable degree polynomials depending on the 
special conditions relative to the number and distribution of control 
points. The study included sixth degree. 
6 Forest Service procedure for stereotriangulation adjustment by 
elevtronic computer by S.E. Webb and O.R. Perry, Photogrammetric 
Engineering, Vol. 25, No. 3, page 404, 1959. 
7 An outline of photogrammetry by K. Schwidefsky, Pitman Publishing 
Corp., p. 272, 1959. 
8 Recent developments in analytic aerial triangulation at the Ordnance 
Survey by D.W.G. Arthur, Photogrammetric Record, Vol. 3, No. 14, page 120, 
1959. 
9rmproved stereotriangulation adjustments with electronic computers by 
A.L. Norwicki and C.J. Born, Photogrammetric Engineering, Vol. 26, No. 4, 
page 599, 1960. 
58 
The unusual terms in the Coast Survey formulas are designed 
to compensate for the local tilts of the strip: otherwise these 
formulas are not greatly different from the others. Justifications 
for their existence is given in the next section. 
It may be appropriate at this stage to state the precision 
toward which this study is directed. Accuracies of a few feet have 
been experienced where the flight altitude is 20,000 feet; fractions 
of a foot are significa~t. Consequently, this program is prepared 
so as to preserve thousandths of a foot for round-off reasons even 
though the small distances are not ordinarily significant in themselves. 
Thus the fine precision being sought causes one to consider carefully 
the type of transformation being applied lest the transformation itself 
add systematic errors due to excessive constraint or relaxation. 
The best root-mean-square accuracy that can be expected using 
the piecemeal, provisional Coast Survey analytic solution is probably 
in the neighbourhood of l/10,000 to 1/20,000 of the flight altitude 
where film is used in the aerial camera; that is, 1 foot if the 
altitude is 10,000 feet. If glass plates are used in the camera, and 
if the results are refined by a subsequent block adjustment technique, 
present results suggest that l/50,000 can be approached; that is, about 
2 1/2 inches if the altitude is 10,000 feet. 
59 
BASIS FOR THE FORMULATION 
Analysis·af the·curve Forms 
The basis for the Coast Survey formulation is essentially 
that of Brandt10 and Price11 and is restated here for the sake of 
completeness. 
Considering the abscissa direction first, the 11 new11 or 
correct value x• (referred to the axis-of-flight coordinate system) 
for a point on the centerline (axis of flight) of a strip of aero-
triangulation is considered to be composed of the 11 old 11 value plus a 
correction ex: 
x• = x + c 
X 
(which also serves to define ex = x• - x}. The correction is expressed 
by means of a polynomial of third degree in terms of the 11 0ld 11 
coordinate: 
x• = x + ax3 + bx2 + ex + f . (6} 
Similarly, the new y-coordinate of a point on the centerline is expressed 
using a quadratic polynomial: 
y• = y + dx2 + ex + g (7} 
also in terms of the abscissa x inasmuch as the magnitude of the 
correction is obviously related to the distance from the beginning of 
the strip. 
Adequate theoretical and operational justification exists for 
assuming that the x and z equations need to be cubic whereas the y-
10 Resume of aerial triangulation adjustment at the Army Map Service by 
R.S. Brandt, Photogrammetric Engineering Vol. 17, No. 4, page 806, 1951. 
11 some analysis and adjustment methods in planimetric aerial triangulation 
by C.W. Price, Photogrammetric Engineering, Vol. 19, No.4, page 627, 
1953. 
60 
equation does not need to be more than quadratic, but it was the 
experience of Coast Survey photogrammetrists based on having performed 
perhaps a hundred or more graphic solutions prior to the use of the 
computer that led to the final conclusions. The x-curve was almost 
never symmetrical (a quadratic curve could by symmetrical) inasmuch as 
the second half invariably had a greater degree of curvature than thefirsthalf. A third degree correction polynomial adequately removed the 
discrepancy whereas the quadratic form left a residual error too large 
to be acceptable, although the situation could not be completely 
explained through a theoretical analysis. However, the y-curve was 
both smaller in magnitude and more nearly symmetrical. The graphic 
z-curves also were perceptably greater than second degree and the 
theoretical reasons seemed to be even more convincing. 
Considering the y-curve for a moment, it is emphasized that 
Equation 7 applies to points on the centerline of the strip (fig. 1). 
The point m on the centerline needs to be corrected by moving it ton, 
and the magnitude of the correction is given by the equation. However, 
the point p on the edge of the strip needs to be corrected in two 
directions: 
(l) one component pq is equal to the correction mn at m, and 
(2) the second component is rq in the x-direction. 
In the right triangle mps, the angle at m is given by the first derivative 
of the equation of they-curve, which is the centerline (Equation 7): 
tan e = (2d) x + e. 
Then 
ps = mp tan e. 
61 
But mp is the ordinate y of the point: 
qr = ps = y[(2d)x + e] . (8) 
The complete x-equation is therefore composed of both equations 6 and 8: 
x• = x + ax3 + bx2 + ex - (2d) xy - ey + f (9) 
where the minus signs derive from the analytic definition of the 
direction of the slope; i.e., the relative direction of rq. 
The complete y-equation is formed through a comparable 
analysis: 
y• = y + 3ax2y + 2bxy + cy + dx2 + ex + g. ( 10) 
Equation 6 depicts a lengthwise stretching or compressing of the strip. 
The term (3ax2y + 2bxy + cy) in Equation 10 is the effect of the local 
stretch or compression in the y-direction both as a function of the 
abscissa x of point in the strip and also as a function of the distance 
y that the point is off the centerline. 
The vertical dimension is explained by a similar analysis. 
The basic equation is the cubic form 
''[' 
l', ·i' 
' ' t., 
I; 1/ ~ ' ' \ ' I} ' '\ II ---~ ·----r-~ { 
:... ' -!...":'r---
"' Mu...:.imum Ordinate, CYBOW 
Fig. 1. -Sketch of the center line of they or azimuth curve illustrating 
the derivation of a component x-correction for a point not on the center 
line. The definition of the term maximum ordinate or CYBOW is also 
indicated. 
z• = z + hx3 + ix2 + jx + kx2y + txy +my+ n 
which can be considered as composed of the sum of the two principal 
geometric parts 
( 11) 
( 12) 
( 13) 
62 
The first part {equation 12) is sometimes called the 11BZ-
curve 11 • This curve is the projection of the center line of the strip 
onto the xz-plane. The second part (Equation 13) has been described 
as "twist" and as "cross tilt ... The latter is considered to be quadratic, 
again based largely on experience gained from the graphic analyses of 
many strips, inasmuch as the graphic curves invariably were not straight 
lines. If they were linear, the effect would resemble a helix of 
constant pitch, like a screw thread, but the quadratic form fits most 
situations more closely. The terms in Equation 13 have as their common 
factor the ordinate y of the point so that the farther the point is off 
the axis of the strip, the greater is the correction. 
T-he Slope Corrections 
If y is factored out of Equation 13, 
(kx2 + tx + m)y . ( 14) 
it is obvious that the parenthetical expression represents the slope of 
the strip perpendicular to the centerline: 
tan w = kx2 + tx + m ( 15) 
Moreover, the first derivative of Equation 12 is the instantaneous slope 
of the BZ-curve in the direction of the strip: 
tan ~ = d/dx (hx3 + ix2 + jx) = 3hx2 + 2fx + j . ( 16) 
Thus equations 15 and 16 depict the slopes of the strip 11 ribbon 11 in the 
x andy directions at any given abscissa x. Then the resultant tilt~ 
(deviation from the vertical) of the normal to the strip is given by 
63 
Fig. 2 - Sketch of the center line vertical BZ-curve illustrating how 
the local inclination of the curve causes a component horizontal 
correction in the x-direction. 
(17) 
which, inasmuch as ~and ware both small angles, is approximated with 
sufficient accuracy for practical operations by 
secT= (1 + tan2 ~ +tan w) 112 ( 18) 
In aerotriangulation, both in instrumental and in the Coast 
Survey analytic systems, the observed or computed coordinates of points 
are related to the initial rectangular axes of the strip at the first 
model or first photograph rather than the curved centerline of the strip 
(fig. 2). Consequently, the base and top of an elevated object have 
different horizontal coordinates. In the figure, the base a has the 
abscissa of point b whereas the top has the abscissa of point c. The 
previous photogrammetric solution yields the abscissa of c and the ele-
vation ~z. which introduce a discrepancy ~x equal to the distance from 
b to c: 
~x = ~z tan ~ = ~z (3hx2 + 2ix + j). ( 19) 
The value is subtracted from the abscissa of c as a correction indicated 
in the first formula of Equation 1. 
In a similar manner, the correction of the ordinate is: 
~Y = ~z tan w = 6z(kx2 + ~x + m) (20) 
64 
as ·indicated in the second formula of Equation 1. 
The z-correct·ion is probably of minor consequence; neverthe-
less, it is also applied. The photogrammetric elevation is too small 
and needs to be increased by multip"lying it by the secant of the 
resultant inclination T of the line perpendicular to the surface of 
the str-ip. The term is applied in the third formula of Equation 1. 
T\!_q__ Pre1 imin?-..r:t3ffine Transformati~-~ 
In Equations 1 it was tactily assumed both that 
(1) the directions of the x, y, z axes were essentially parallel to the 
x', y', z' axes, and also that 
(2) the x-axis represented the centerline or axis of flight of the 
photogrammetric strip. 
Both of these cond-itions are violated in practice; consequent-ly, two 
preliminary affine transformat·ions are utilized for rotation, trans-
lation, and scale change. Inasmuch as these conditions need not be 
exactly adhered to, unique transformations are utilized so as to impose 
asfew fixed conditions as possible and to simplify the computations 
for determ·ining the constants of the transformation equations. 
Perhaps an exp'lanation is in order as to the reasons for 
assuming that these conditions are necessary. The systematic errors 
depicted by the polynomials of Equation 1 are direction-sensitive 
inasmuch as they are propagated as functions of the length of the strip, 
or the number of photographs in the strip, or simply the abscissa of a 
po·int in the strip. However, the photogrammetric direction of the strip 
(axis of flight) is not known with sufficient accuracy until after 
the strip has been aerotriangulated, at which time the easiest way to 
obtain the desired coor·dinates is to transform them by means of a 
computer rather than to reobserve them. 
65 
Secondly, whereas the photogrammetric strip (model) coordinates 
may progress in any direction of the compass, the ground coordinates are 
oriented with +X eastward. But the X-coordinates must also be 
reoriented into the axis-of-flight direction, which possibly would 
be unnecessary if all strips were flown north-south or east-west. Again, 
it is fairly easy to rotate, scale and translate the ground coordinate 
system into the axis-of-flight system with an electronic computer. 
Finally, after the correction Equations 1 have all been 
applied to the coordinates of a point, it is necessary to convert the 
coordinates back into the ground system by applying the inverse of the 
second affine transformation above so that the resulting coordinates 
are meaningful and useful in surveyingand mapping work. 
Model Coordinates to Axis-of-Flight System 
By 11 model 11 coordinates is meant the form of the data from a 
stereoplanigraph bridge. A comparable form results from the preliminary 
computer solution of the Coast Survey analytic aerotriangulation. 
Let the model coordinates of a point near the center of the 
initial model be x1, y1, and near the center of the terminal model be 
x2, y2. The axis of flight is arbitrarily defined as passing through 
these two points. This axis of flight is to be the new x-axis: the 
new ordinates of both the above points are therefore zeros. The origin 
of the axis of flight is defined as midway between these initial and 
terminal points in order to reduce the numerical magnitude of the x-
coordinates, which has added importance inasmuch as the x-coordinates 
_are squared and cubed as indicated in Equation 1. The distanceD 
66 
between the initial and terminal points is given by analytic geometry 
to be 
Consequently, the coordinates of the initial and terminal points in 
the axis-of-flight system are 
x• - -1 l/2 D, yl = 0 
x2 = + l/2 D, Y2 = 0 
(21) 
The axis-of-flight coordinates of other points can be computed 
using the following set of affine transformation formulas (which comprise 
a special form of the more general Equation 25 discussed later): 
x• = a1x - b1y + c1 
(22) 
a, =' -flx/D 
b, = lly/D 
c, = -a1 x1 + blyl l/20 (23) 
dl = -a1 x1 - alyl 
Equations 22 constitute simply a rotation and translation of 
the 11 model 11 coordinates into the 11 axis-of-flight 11 coordinates maintain-
ing the same original model scale. The coefficients a1 .•... d1 are 
the constants for the transformation: their values are determined once 
only. 
67 
Ground Horizontal Coordinates Into the Axis-of-Flight System, and the 
Inverse 
The coordinates of horizontal ground control stations also 
need to be transformed into the same axis-of-flight system. The 
transformation is based on only two of the stations, one near the 
initial end of the strip and one near the terminal end. In the 
program, the coordinates of the two stations are listed as the first 
and the last ones used in the adjustment. Two sets of coordinates are 
given for each point: one set is in the form of model coordinates 
x1 ... y2 and the other set in the ground survey system as x1 ... v2. 
The first step is to transform the model values x1, etc., 
into the axis-of-flight system by applying Equations 22 and then 
applying the slope corrections as indicated by Equations 19 and 20. 
If xl ... Y2 are the axis-of-flight coordinates of the initial and 
terminal control stations, the following differences can be expressed: 
(24) 
(~X is called OGX in the Fortran program Statements 12+4 and 80+3). The 
square of the axis-of-flight distance o2 between the terminal model 
control points is 
02 = ~x·2 + ~y·2 . 
The same type of affine transformations as Equation 22 is 
applied to the axis-of-flight coordinates to convert them into the 
ground system of coordinates (Statements 70+1 and 70+2 in the Fortran 
program) which is applied to all new (bridge) points as the last stage of 
the computation after implementing the corrections of Equation 1: 
68 
(25) 
If the values of xl' etc., and x1, etc. are substituted into Equations 
25, one obtains four simultaneous linear equations in which four constant 
coefficients a20 •.... d20 are the only unknowns. The solution of the 
equations gives 
a20 = (~X . ~x· + ~y • ~y')/02 
b20 = (~Y . ~x· - ~x . ~y')/02 
c20 = Xl - a20 xl + b2oYl 
Equations 25 embody a change in scale in addition to rotation and 
translation. 
The inverse form of Equations 25 shows the corresponding 
transformation of ground coordinates into axis-of-flight values: 
x' = a21X + b21Y- c21 
y' = -b2lx + a21Y- d21 
The values of the four new constants can be computed from those of 
Equations 25 by applying the following relations: 
2 2 d '* = 1 I ( a20 + b20 ) 
a21 = a20 d* 
b21 = b20 d* 
c21 = (a20 c20 + b20 d20)d* 
d21 = (a20 d20 - b20 c20)d* 
(26) 
(27) 
(28) 
The evaluation of all eight of the constants of Equations 25 and 27 are 
indicated after Statement 80 of the Fortran program. 
69 
It is noted that (a~0 + b~0 ) 112 expresses the scale change 
included in the transformation. 
It should be noted that the application of transformation 
Equations 27 is such that no discrepancy ex or cy exists or remains at 
the two end horizontal control points on which the transformation is 
based. Consequently, if the two polynomial curves are plotted, they 
will both cross the zero line, or x-axis, at the location of these 
two control stations. (Fig. 1) 
Normally each curve describes a sweeping arc joining these 
two end points. Midway between the end points they-curve will be at 
its greatest distance from the x-axis, indicating that the correction 
is maximum. One is assured of this feature because the equation is 
quadratic and is symmetrical with respect to they-axis. Moreover 
Equations 22 have already been applied so that the origin of the 
observed coordinate system is at the center of the strip. Consequently 
the maximum height of the curve is where it crosses the y-axis which 
is where x = 0. If in Equation 10 x is set equal to zero, then the 
height of the curve is given simply by cy = g. The value g may be 
called the 11 maximum ordinate 11 which is symbolized in the program as 
CYBOW. This quantity is a diagnostic term whose magnitude as deter-
mined through experience normally does not exceed certain practical 
limits unless a blunder occurs in some of the data. 
Similarly, from Equation 9 the CXBOW is ex = f. Inasmuch as 
the x-curve is cubic, it is not symmetrical and the value f is not 
the maximum. Nevertheless, f is sufficiently near the maximum that it 
still is useful for detecting a gross blunder. 
70 
Preliminary Vertical Affine Transform 
The value of z is derived from the photogrammetric analysis 
and its exact relation to the ground system of measurement is not known 
until later. The following transformation and its inverse relating the 
two systems are stated and explained: 
z' = z + Z/s 0 
z = s(~' - ~ } 0 . 
(29) 
(In Fortran notation, z0 is designated as EINDEX. These equations occur 
in the Fortran program as Statements 31+1 and 70+3}. The instrument or 
photogrammetric model elevation of a point in the axis-of-flight system 
is symbolized as z' whereas Z refers to the ground elevation, s is· the 
horizontal (as well as vertical} scale factor that relates the magnitudes 
of z' and Z, and z0 is an index elevation, or translation term. Thus 
Equation 29 implies simply dilation (scale) and translation. The scale 
factor is determined after Equation 24 above: 
(30} 
(Statement 12+5 in the Fortran program). Obviously the scale factor 
is normally a number greater than 1, such as 10,000, and is the number 
for multiplying instrument or model dimensions to convert them into 
equivalent ground dimensions. 
The index elevation z0 is defined as an average value for a 
list of control points: 
71 
(Fortran statement 14+2). Thus the index is the difference between 
the average of the instrument elevations and the average of the corres-
ponding ground elevations that have been scaled into the instrument 
units. The index serves not only as a translational element between 
the two systems, but also confines the adjustment to the actual zone 
of elevations rather than simply applying the adjustment to a set of 
abstract numbers which in practice would each ordinarily be of 
considerably greater magnitude than its corresponding elevation. 
The Solution of Simultaneous Equations